Make the VSCode extension compatible with https://vscode.dev/ and https://github.dev/

Is your feature request related to a problem? Please describe.
Developers are not able to use this extension with https://vscode.dev/ and https://github.dev/

Describe the solution you'd like:
Make the VSCode extension compatible with https://vscode.dev/ and https://github.dev/
@hbenl @rugk @Misiur @jonfen @ilikesortedarrays @jnachtigall
Please consider this as DX improvement and give some priority for it.
Thanks

Describe alternatives you've considered:
No alternatives available.

Additional context:
https://aka.ms/vscode-web-extensions-guide
https://code.visualstudio.com/api/extension-guides/web-extensions

Screenshot of extension showing not compatible with web version.